Does there exist an $n \times n$ matrix $A$ which satisfies the following?
- $A \succeq 0$ ($A$ is positive semidefinite)
- $\sum_i A_{ii} = 0$ (trace of $A$ is zero)
- $A \neq 0$ ($A$ is not the zero matrix)
I know that there exist no such matrices which are symmetric, but I am not sure how to prove that no matrices satisfy the above for the case of non-symmetric matrices, or to otherwise find a counterexample.
Best Answer
Okay actually I asked this question way too hastily...
$A = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} $ is positive semidefinite, nonzero, and has zero trace.